Mixed Control for Discrete-time Systems via Convex ... - Deep Blue

Automatica, Vol. 29, No. 1, pp. 57-70, 1993

0005-1098193 $6.00 + 0.00 ~) 1992 Pergamon Pre'~ Ltd

Printed in Great Britain.

Mixed

Control for Discrete-time Systems via Convex Optimization*t ISAAC KAMINER,* PRAMOD P. KHARGONEKAR~: and MARIO A. ROTEA§

A mixed ~ 2 / ~ problem for discrete-time systems is solved by converting it into a convex optimization problem over a finite-dimensional space. Key Words--Robust control; multiobjective control; convex programming.

designer understand how the various competing objectives conflict with each other. From this point of view, one should postulate the controller synthesis problem as the problem of studying tradeoffs among competing objectives. For a more detailed discussion of multiobjective controller synthesis as well as additional references, see Boyd and Barratt (1990), Dorato (1991), Khargonekar and Rotea (1991b), and Rotea (1990). The subject of this paper is a certain constrained optimal controller synthesis problem--the so-called mixed gz/~® synthesis problem. Mixed g 2 / ~ problems can be motivated in many different ways. As a matter of fact, there are many different mixed g 2 / ~ problems. These problems are one way of analytically formulating the issue of tradeoffs in control system synthesis. To give a brief description of the various mixed ~/f-2/~ problems, consider the feedback system shown in Fig. 1. Let Tz,w,, i = 0, 1, denote the closed loop transfer matrix from the exogenous input w~ to the controlled output z~. One mixed ~g2/~® problem is to find an internally stabilizing controller ca which minimizes IITzo~0112 subject to the constraint IITz,w,ll~ v ( ~ ) can be obtained by solving a finite-dimensional convex programming problem over a bounded set of real matrices. In the output-feedback case, it is shown that the mixed ~ / ~ control problem can be reduced to a full-information feedback problem for an auxiliary plant, which is obtained from the given plant by solving an (~® filtering) algebraic Rieeati equation. Thus, the output-feedback problem can be reduced to a finite-dimensional convex programming problem over a set of real matrices. It is shown that the output-feedback controllers can always be chosen to have a structure similar to that of the standard ~® central controller. This implies that the order of (nearly) optimal output-feedback controllers need not exceed that of the generalized plant. While the approach taken here is somewhat similar to the approach of Boyd and Barratt (1990), in that they also reduce such controller synthesis problems to convex optimization problems, there are significant differences between our results and those of Boyd and Barratt (1990). In particular, we reduce the mixed ~-2/~® controller synthesis problem to a convex optimization problem over a bounded subset of q x n and n x n symmetric real matrices, where q and n are, respectively the control input and the state dimensions. We accomplish this reduction of the problem without finite-dimensional approximations of the set of stabilizing controllers or frequency discretizations. Consequently, a solution to our convex programming problem is a global solution to the mixed ~ / ~ ® synthesis problem. This is considered to be an important contribution of our work. By comparison, the results of Boyd and Barratt (1990) applied to the present problem would reduce it to a convex optimization problem over the infinite-dimensional space of stable transfer functions. Next, we briefly introduce notation used in this paper. The symbol O denotes the empty set. Given a real matrix A, IIAII denotes its maximum singular value, tr (A) denotes its trace, and A' its transpose. We will say that a square matrix A is asymptotically stable if all its eigenvalues are inside the open unit disk. For A and B real symmetric matrices, A > B (respectively A-> B) iff the difference A - B is

59

positive-definite (respectively, positive-semidefinite). Linear time-invariant systems described by state space equations and are denoted by the script symbols, whereas the corresponding transfer matrices denoted by italics. For example, c~ denotes a system with transfer function G. The Hardy spaces ~-2 and ~¢®consist of matrix valued functions that are square integrable and essentially bounded, respectively, on the unit circle with analytic extension outside of the unit circle. The norms on these spaces are defined in the usual way. 2. THE MIXED ~ / ~ ® PERFORMANCE MEASURE

In this section, we will define the mixed ~ z / ~ performance measure. This will then be used in setting up the controller synthesis problem in the next section. Let us begin by considering a finitedimensional linear time-invariant discrete-time system 3" as shown in Fig. 2. Suppose that 3-is internally stable with the discrete-time state-space model: f (ax)(k) := x ( k + 1) = Fx(k) + Gw(k), er:= o(k) = noX(k) + Jow(k), |

I. zl(k) = H,x(k) + J1w(k), (1) where the matrices F, G, Hi and Ji are real and of compatible dimensions, and F has all eigenvalues in the open unit disk. (In the sequel, we will not show the time variable k explicitly in system equations.) Let Tzw

LTz,wJ

denote the transfer matrix from w to z = zi]'. Let L¢ denote the controllability gramian of the pair (F, G), i.e. Lc is the unique solution of the Lyapunov equation FLcF' + GG' = Lc.

(2)

Then, as is well known,

IIT o.Jl = tr (HoLoH + Jj ). Let y > 0 be given, and consider the transfer matrix Tz,w. In this paper, we will be interested in the ~,~ norm bound IITz,wll~ 0 such that

M(Y) := 721 -

J,J', - H, YH[ > O, and

R(Y) := FYF' - Y + (FYH'~ + GJ~)M-' x(H~YF'+J~G')+GG' 0 such that

M(Y) :=

y21 - JlJ; - 1-11YH; > 0, and

R ( Y ) := FYF' - r + (FYH~ + GJ;)M -~ x(HIYF' +J,G')+GG'=O,

(6)

and F + (FYH; + GJ;)M-~H~ is asymptotically stable. (In fact, Y satisfying the above conditions is unique.) Moreover, if 12 denotes a solution to either (4) or (5), then Y-< I2.

Proof. The equivalence of items 1 and 5 can be found in Molinari (1975). The equivalence of items 1 and 4 follows from the equivalence of items 1 and 5 and a standard small perturbation argument. The equivalence of items 3 and 4 follows from simple algebraic manipulations and the Schur complement formula. Setting Y := ( p , p ) - i yields 2 ~ 3 , and setting P = y - m gives 3 ~ 2 . Finally, in item 5, the inequality Y-< I7" follows from Ran and Vreugdenhil (1988). • Now suppose IIT=,~II~ < ~. Let Y denote the unique real symmetric matrix that satisfies condition 5 in Theorem 2.1. Then, from the definition of the controllability gramian L~ and Theorem 2.1, it follows that 0-< L~ -< Y.

(7)

Note that this is the best possible upper bound

for the controllability gramian that may be defined in terms of the solutions to the various quadratic matrix inequalities in Theorem 2.1. Thus, t ~ t IIT~.,Ih2 = tr (HoLcHoI + JoJo) - tr (H0 YHo + JoJ~).

The above inequality motivates the following definition of the mixed ~-,2/~ performance measure (or cost) J(Tzw) for the linear time-invariant system 3-: J(T~w) := tr (HoYH~ + JoJ~).

(8)

The performance measure defined in (8) is the same as the one considered by Mustafa and Bernstein (1991), Bambang et al. (1990), and Haddad et al. (1991). (More precisely, this performance measure is one of the costs considered in Mustafa and Bernstein (1991).) It is easily seen that J(Tzw) is only a function of the transfer matrix T~, and does not depend on the choice of realization, as long as such a realization is internally stable. This justifies our notation. Also 11T~,wl[2 0 such that: (1) M : = I - J 1 J ~ - H 1 Y H ~ > O (2) R(Y) := FYF' - Y + (FYH; + G J;) x M - I ( H I Y F ' + J I G ' ) + GG' 0 and R~ 0 by taking Schur complement. Simple algebraic manipulations show that the "1-1 block" of R(Y) satisfies

--
0, (20), (21) and the implication 4 ::> 3 of Theorem 2.1 imply that I:1 satisfies

Theorem 4.2. Consider the state-feedback plant ~: defined in (13). Then

R, = FraY,F " - }'1 + OmG'm + BzC~QC'cB~ + (FmY~n~m + Grj~m + B2CcQC'D~2)M -~ x (H~mYIF" +J~mB~m + D~2CcQC'B~) < 0 , (20) where M = I - JlmJ~m HlmYiHim -DI2C~QC~D~2>O. -

-

=/=

0,

where ~(~ds:) and ~t~.,,(~d,i) are as in (9) and (11), respectively. In this case =

It follows that

..[_[jGl mm][G~J,lm]< [Y1 ~1, (22) since Q > 0. Now using (22) and Y1> 0, a simple Lyapunov argument shows that Fm is a stable matrix. Further, from implication 3 => 1 in Theorem 2.1, it follows that IITz,~(G~,K)II® v,,(~/i), there exists a triple (W, Y, K2) • di~(~li) such that the static full-information controller K : = [WY-*

K2],

satisfies • M.,,(~da)

and

m'

[B,

c; 1

/7:= B~ D~2J'

w']',

Du

D12.1

g:=[l

By Proposition E.7.f in Marshall and Olkin (1979) the mappings (ff,,y)_.,/Cff, y - l f f - , p , , and/~--* GKR'G', are convex on their domains (here Y = Y' >0). Since the maps (W, Y)--* [Y' W']' and K2--" [I K~]' are alfine linear, the convexity of L follows. Finally, the convexity of ~(~t#) follows from the convexity of L. •

J(Ga, K) < o~.

This result is a direct and straightforward generalization of Theorem 4.2 of Khargonekar and Rotea (1991a). Proof is omitted. In the remainder of this section we will show that the optimization problem defined in (27) is convex.

Lemma 4.6. Consider the set ~ ( ~ ) defined in (26). Assume that /)12 has full column rank. Suppose that for all z inside the open unit disc, the system matrix ZI -- a -C 1

B2 1, O12_1

~ 2 / ~ control for discrete-time systems

65

has full column rank. Then the set tb(~d~) is bounded.

This contradicts (29), and therefore

Proof. We need to show that there exist positive constants ml, m2, m3 < ~ such that

Using (32) and taking Schur complement of the (2, 2) block in (28), it follows that

IIKzll-< ms.

Let (W, Y, KE)eO(~g). From definitions of • (~) and L(W, Y, K2), it follows that D12K2K~D~2 < I, which implies that I[D12K2][< 1. Since DiE is full column rank there exists a constant ms < oo, such that IIKz[I ~ ms < ~. Now define the matrices

T:=[Io

-B2XD'12 ] I-D12XD~2J'

,4 := A - B2XD~2C1,

IICIY + D12WI[ < lIVId. Since [[Yl[-O.

•

5. OUTPUT-FEEDBACK CASE

In this section we will solve the synthesis problem defined in Section 3 for the outputfeedback case. We will show that the problem can be reduced to solving one algebraic Riccati equation, and a convex optimization problem similar to the one in Section 4. In the following subsection we introduce a technical result that will be needed in order to prove the main theorem of this section.

5.1. Preliminaries The next result is an extension of Redheffer's lemma (see, for example, Iglesias and Glover (1991) and Stoorvogel (1990)) to the mixed St-2/$t~ performance measure for discrete-time systems. Consider the feedback interconnection of Fig. W ~)

(31)

Premultiplying (31) by y' and postmultiplying it by y yields

y ' ( i - C1YC;)y >0. t

IIWll~m2 0 be given. Solve the convex program (27), corresponding to c~,(Q), to compute a full-information gain K = [K1 K2] • M~,~(~3~(Q)) such that J(Ga(Q), K)